Interconnected systems are an important class of mathematical models, as they allow for the construction of complex, hierarchical, multiphysics, and multiscale models by the interconnection of simpler subsystems. Lagrange--Dirac mechanical systems provide a broad category of mathematical models that are closed under interconnection, and in this paper, we develop a framework for the interconnection of discrete Lagrange--Dirac mechanical systems, with a view towards constructing geometric structure-preserving discretizations of interconnected systems. This work builds on previous work on the interconnection of continuous Lagrange--Dirac systems (Jacobs and Yoshimura 2014) and discrete Dirac variational integrators (Leok and Ohsawa 2011). We test our results by simulating some of the continuous examples given in Jacobs and Yoshimura 2014.
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange-d'Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators…
Geometric integrators for higher-order variational systems and their application to optimal control
University of Michigan（密歇根大学）；Universidad Nacional del Sur（国立苏尔大学）；Instituto de Matemática Bahía Blanca（巴亚布兰卡数学研究所）；CONICET（阿根廷国家科学研究委员会）；Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)（数学科学研究所（西班牙高等科学研究理事会-马德里自治大学-马德里理工大学-马德里 Complutense 大学））
We develop an explicit, second-order, variational time integrator for full body dynamics that preserves the momenta of the continuous dynamics, such as linear and angular momenta, and exhibits near-conservation of total energy over exponentially long times. In order to achieve these properties, we parametrize the space of rotations using exponential local coordinates represented by a rescaled form of the Rodrigues rotation vector and we systematically derive the time integrator from a discrete Lagrangian function that yields discrete Euler-Lagrange equations amenable to explicit, closed-form s…
Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this paper we present two ways to construct r-adaptive vari…
In this paper, we present two Hermite polynomial based approaches to derive one-step numerical integrators for mechanical systems. These methods are based on discretizing the configuration using Hermite polynomials which leads to numerical trajectories continuous in both configuration and velocity. First, we incorporate Hermite polynomials for time-discretization and derive one-step variational methods by discretizing the Lagrange-d'Alembert principle over a single time step. Second, we present the Galerkin approach to derive one-step numerical integrators by setting the weighted average of th…
